Having moved through the preceding compactness ideas, we now study compactness in discrete and Euclidean spaces. The focus keyword for this lesson is compactness in discrete and Euclidean spaces. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::theorem[Compact Subsets of a Discrete Metric Space] In a discrete metric space, a subset is compact if and only if it is finite. ::: :::proof Given that the metric is discrete. To prove the criterion. Finite sets are compact. Conversely, the singleton cover of a compact subset has a finite subcover, so the subset must be finite. $\square$ ::: :::theorem[Heine-Borel Theorem in Euclidean Space] A subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded. ::: :::proof Given that $Y\subseteq\mathbb{R}^n$. To prove Heine-Borel. Compact subsets are closed and bounded. Conversely, a closed bounded set lies in a compact cube $[-M,M]^n$ and is closed inside that cube. Hence it is compact. $\square$ ::: :::corollary Under any standard equivalent Euclidean metric, compactness in $\mathbb{R}^n$ is the same as closedness and boundedness. ::: :::exercise [1] State the main definition or theorem from this lesson. [2] Give one example where the hypothesis is used. [3] Explain one common mistake related to this topic. ::: :::answer [1] See the first formal block of the lesson. [2] The examples in the lesson provide standard choices. [3] A common mistake is to apply a Euclidean compactness rule in an arbitrary metric space. ::: :::faq[Frequently Asked Questions] Q: Why is the metric important? A: Compactness, boundedness, convergence, and openness are all interpreted using the chosen metric. Q: What should students check first? A: Identify the space, the metric, and the exact compactness criterion being used. ::: :::call-to-action[Revise Compactness] subtitle: Review the definitions, proofs, and examples before moving to the next compactness idea. button: First Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/open-covers-and-compactness button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/compactness-criteria-in-sequence-spaces :::